Oberwolfach Preprints (OWP)

Permanent URI for this collectionhttps://oa.tib.eu/renate/handle/123456789/14626

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Parabolic Normalizers in Finite Coxeter Groups as Subdirect Products
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Douglass, J. Matthew; Pfeiffer, Götz; Röhrle, Gerhard
We revisit the structure of the normalizer $N_W(P)$ of a parabolic subgroup $P$ in a finite Coxeter group $W$, originally described by Howlett. Building on Howlett's Lemma, which provides canonical complements for reflection subgroups, and inspired by a recent construction of Serre for involution centralizers, we refine this understanding by interpreting $N_W(P)$ as a subdirect product via Goursat's Lemma. Central to our approach is a Galois connection on the lattice of parabolic subgroups, which leads to a new decomposition \begin{align*} N_W(P) \cong (P \times Q) \rtimes ((A \times B) \rtimes C)\text, \end{align*} where each subgroup reflects a structural feature of the ambient Coxeter system. This perspective yields a more symmetric description of $N_W(P)$, organized around naturally associated reflection subgroups on mutually orthogonal subspaces of the reflection representation of $W$. Our analysis provides new conceptual clarity and includes a case-by-case classification for all irreducible finite Coxeter groups.
The $q$-Deformed Random-to-Random Family in the Hecke Algebra
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Brauner, Sarah; Commins, Patricia; Grinberg, Darij; Saliola, Franco
We generalize Reiner-Saliola-Welker's well-known but mysterious family of $k$-random-to-random shuffles from Markov chains on symmetric groups to Markov chains on the Type-$A$ Iwahori-Hecke algebras. We prove that the family of operators pairwise commutes and has eigenvalues that are polynomials in $q$ with non-negative integer coefficients. Our work generalizes work of Reiner-Saliola-Welker and Lafrenière for the symmetric group, and simplifies all known proofs in this case.
Scalar Curvature in Dimension 4
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Deng, Jialong
We prove that every locally conformally flat metric on a closed, oriented hyperbolic $4$-manifold with scalar curvature bounded below by $-12$ satisfies Schoen’s conjecture. We also classify all closed Riemannian $4$-manifolds of positive scalar curvature that arise as total spaces of fibre bundles. For a closed locally conformally flat manifold $(M^4,g)$ with scalar-flat and $\pi_2(M^4) \neq 0$, we show that the universal Riemannian cover $(\widetilde{M},\tilde{g})$ is homothetic to the standard product $\mathbb{H}^2 \times \mathbb{S}^2$. This affirmatively answers a question of N. H. Noronha.
Inversion of the Unbounded Finite Hilbert Transform on $L^1$
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Curbera, Guillermo P.; Okada, Susumu; Ricker, Werner J.
The finite Hilbert transform $T$ is a classical singular integral operator with its roots in aerodynamics, elasticity theory and image reconstruction. The setting has always been to consider $T$ as acting in those rearrangement invariant spaces $X$ over (−1, 1) which $T$ maps boundedly into itself (e.g., $L^p$ for 1 < $p$ < ∞), a setting which excludes $L^1$. Our aim is to go beyond boundedness and to address the case $X$ = $L^1$. For this, we need to consider $T$ as an unbounded operator on $L^1$. Is there a “suitable” domain for $T$? Yes. Remarkably, for $T$ acting on this domain, we prove a full inversion theorem, together with refined versions of both the Parseval and Poincaré-Bertrand formulae, which are crucial results needed for the proof. This domain, a somewhat unusual space, turns out to be a rather extensive subspace of $L^1$, fails to be an ideal and properly contains the Zygmund space $L$log$L$ (which is the largest ideal of functions that $T$ maps boundedly into $L^1$).
Some Notes on Pontryagin Duality of Abelian Topological Groups
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Hofmann, Karl Heinrich; Kramer, Linus
We consider several questions related to Pontryagin duality in the category of abelian pro-Lie groups.
Renormalisation of Singular SPDEs with Correlated Coefficients
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Clozeau, Nicolas; Singh, Harprit
We show local well-posedness of the g-PAM and the $\phi^{K+1}_2$-equation for $K\geq 1$ on the two-dimensional torus when the coefficient field is random and correlated to the driving noise. In the setting considered here, even when the model in the sense of [Hai14] is stationary, naive use of renormalisation constants in general leads to variance blow-up. Instead, we prove convergence of renormalised models choosing random renormalisation functions analogous to the deterministic variable coefficient setting. The main technical contribution are stochastic estimates on the model in this correlated setting which are obtained by a combination of heat kernel asymptotics, Gaussian integration by parts formulae and Hairer-Quastel type bounds [HQ18].
Generalized Bose-Einstein Condensation in the Kac-Luttinger Model
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Boccato, Chiara; Kerner, Joachim; Pechmann, Maximilian; Spitzer, Wolfgang
In this article, we prove generalized Bose–Einstein condensation (BEC) at zero temperature in the random Kac–Luttinger model for repulsive two-particle interactions that are scaled suitably in the limit of large volume. Compared to previous works, by proving generalized condensation rather than the macroscopic occupation of finitely many single-particle states (type-I BEC), we can allow for stronger two-particle interactions. We discuss implications of the result which include a possible transition in the type of the condensation.
When Alcoved Polytopes Add
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Early, Nick; Kühne, Lukas; Monin, Leonid
Alcoved polytopes are characterized by the property that all facet normal directions are parallel to the roots $e_i-e_j$. Unlike other prominent families of polytopes, like generalized permutahedra, alcoved polytopes are not closed under Minkowski sums. We nonetheless show that the Minkowski sum of a collection of alcoved polytopes is alcoved if and only if each pairwise sum is alcoved. This implies that the type fan of alcoved polytopes is determined by its two-dimensional cones. Moreover, we provide a complete characterization of when the Minkowski sum of alcoved simplices is again alcoved via a graphical criterion on pairs of ordered set partitions. Our characterization reduces to checking conditions on restricted partitions of length at most six. In particular, we show how the Minkowski sum decompositions of the two most well-known families of alcoved polytopes, the associahedron and the cyclohedron, fit in our framework. Additionally, inspired by the physical construction of one-loop scattering amplitudes, we present a new infinite family of alcoved polytopes, called $\widehat{D}_n$ polytopes. We conclude by drawing a connection to matroidal blade arrangements and the Dressian.
Spectral Regularity and Defects for the Kohmoto Model
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Beckus, Siegfried; Bellissard, Jean; Thomas, Yannik
We study the Kohmoto model including Sturmian Hamiltonians and the associated Kohmoto butterfly. We prove spectral estimates for the operators using Farey numbers. In addition, we determine the impurities at rational rotations leading to the spectral defects in the Kohmoto butterfly. Our results are similar to the ones obtained for the Almost-Mathieu operator and the associated Hofstadter butterfly.
Domain-Scaled Regular Variation: Mathematical Foundations for a New Tail Process
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Strokorb, Kirstin; Oesting, Marco; De Fondeville, Raphaël
Threshold exceedances of stochastic processes in space and time often appear to be more localized the more extreme they are. While classical regularly varying stochastic processes cannot model this effect, we introduce an adapted version of regular variation, where a suitable domain-scaling can be incorporated to accommodate this behaviour. Our theory is inspired by the triangular array convergence of domain-scaled maxima of Gaussian processes to a Brown-Resnick process and turns out to be natural in this context. We study key properties of the resulting tail process and demonstrate its ability to approximate conditional exceedance probabilities of Gaussian processes. Mathematical convenience arises from the recently rediscovered concept of vague convergence based on boundedness.
Renormalised Models for Variable Coefficient Singular SPDEs
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Broux, Lucas; Singh, Harprit; Steele, Rhys
In this work we prove convergence of renormalised models in the framework of regularity structures [Hai14] for a wide class of variable coefficient singular SPDEs in their full subcritical regimes. In particular, we provide for the first time an extension of the main results of [CH16, HS24, BH23] beyond the translation invariant setting. In the non-translation invariant setting, it is necessary to introduce renormalisation functions rather than renormalisation constants. We show that under a very general assumption, which we prove covers the case of second order parabolic operators, these renormalisation functions can be chosen to be local in the sense that their space-time dependence enters only through a finite order jet of the coefficient field of the differential operator at the given space-time point. Furthermore we show that the models we construct depend continuously on the coefficient field.
Quadratically Enriched Plane Curve Counting via Tropical Geometry
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Jaramillo Puentes, Andrés; Markwig, Hannah; Pauli, Sabrina; Röhrle, Felix
We thank Jesse Pajwani for pointing out that identities in the Grothendieck-Witt ring can be checked on multiquadratic finite ´etale algebras and for his help with computations. We thank Erwan Brugall´e, Andreas Gross, Marc Levine, Dhruv Ranganathan and Kirsten Wickelgren for useful discussions. The first, second and fourth author acknowledge support by DFG-grant MA 4797/9-1. The third author acknowledges support by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Collaborative Research Centre TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124. The first author thanks the Universität Duisburg-Essen and the Università degli Studi di Napoli Federico II for support. Part of this work was completed while the authors stayed as Research Fellows at the Mathematisches Forschungsinstitut Oberwolfach in March 2024. We thank the institute for hosting us and for providing ideal working conditions.
On Angular Momentum Algebras and their Relations
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Calvert, Kieran; de Martino, Marcelo; Oste, Roy
In this paper, we study the centraliser of $\mathfrak{osp}(1|2)$, denoted the total angular momentum algebra (TAMA), in the Weyl Clifford algebra. The TAMA extends the angular momentum algebra (AMA), which arises as the centraliser of $\mathfrak{sl}(2)$ and admits a diagrammatic presentation via the crossing relation described by Feigin and Hakobyan. Using Young symmetrisers we construct an analogue relation for the even subalgebra of the TAMA. We prove that for rank $4$ and $5$ these relations generate a presentation for the even subalgebra of the TAMA.
On the Halpern Method with Adaptive Anchoring Parameters
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Pinto, Pedro; Pischke, Nicholas
We establish the convergence of a speed-up version of the Halpern iteration with adaptive anchoring parameters in the general geodesic setting of Hadamard spaces, generalizing a recent result by He, Xu, Dong and Mei from a linear to a nonlinear setting. In particular, our results extend the fast rates of asymptotic regularity obtained by these authors for the first time to a nonlinear setting. Our approach relies on a quantitative study of these previous results in the linear setting, combined with certain optimizations and an elimination of the weak compactness arguments employed crucially in the linear setting, which not only allows for the lift of the result to a nonlinear setting but also streamlines the previous convergence analysis considerably. This work is set in the context of recent developments in proof mining, and as byproduct of our approach, we further obtain quantitative information in the form of highly uniform rates of metastability of low complexity, which are new already in the context of Hilbert spaces.
A CFSG-Free Explicit Jordan’s Theorem over Arbitrary Fields
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Bajpai, Jitendra; Dona, Daniele
We prove a version of Jordan's classification theorem for finite subgroups of $\mathrm{GL}_{n}(K)$ that is at the same time quantitatively explicit, CFSG-free, and valid for arbitrary $K$. This is the first proof to satisfy all three properties at once. Our overall strategy follows Larsen and Pink [24], with explicit computations based on techniques developed by the authors and Helfgott [2, 3], particularly in relation to dimensional estimates.
On the Complexity of Epimorphism Testing with Virtually Abelian Targets
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2025) Elder, Murray; Shen, Jerry; Weiss, Armin
Friedl and Löh (2021, Confl. Math.) prove that testing whether or not there is an epimorphism from a finitely presented group to the direct product of an abelian and a finite group, or to a virtually cyclic group, is decidable. Here we prove that these problems are NP-complete. In addition we show that testing epimorphism is NP-complete when the target is a restricted type of semi-direct product of a finitely generated free abelian group and a finite group, thus extending the class of virtually abelian target groups for which decidability of epimorphism is known. We also consider epimorphism from a finitely presented group to a fixed finite group. We show the epimorphism problem is NP-complete when the target is a dihedral group of order that is not a power of 2, complementing the work on Kuperberg and Samperton (2018, Geom. Topol.) who showed the same result when the target is non-Abelian finite simple.
Alternating Snake Modules and a Determinantal Formula
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Brito, Matheus; Chari, Vyjayanthi
We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and the modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to classical questions in the category $\mathcal{ O}(\mathfrak{gl}_r)$. Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights $\mu$ for which the non-zero Kazhdan-Lusztig coefficients $c_{\mu, \nu}$ are $\pm 1$.
The Congruence Properties of Romik’s Sequence of Taylor Coefficients of Jacobi’s Theta Function θ3
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Krattenthaler, Christian; Müller, Thomas W.
In [Ramanujan J. 52 (2020), 275-290], Romik considered the Taylor expansion of Jacobi's theta function θ3(q) at q=e−π and encoded it in an integer sequence (d(n))n≥0 for which he provided a recursive procedure to compute the terms of the sequence. He observed intriguing behaviour of d(n) modulo primes and prime powers. Here we prove (1) that d(n) eventually vanishes modulo any prime power pe with p≡3 (mod 4), (2) that d(n) is eventually periodic modulo any prime power pe with p≡1 (mod 4), and (3) that d(n) is purely periodic modulo any 2-power 2e. Our results also provide more detailed information on period length, respectively from when on the sequence vanishes or becomes periodic. The corresponding bounds may not be optimal though, as computer data suggest. Our approach shows that the above congruence properties hold at a much finer, polynomial level.
Diameter and Connectivity of Finite Simple Graphs II
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Hibi, Takayuki; Saeedi Madani, Sara
Let $G$ be a finite simple non-complete connected graph on $[n] = \{1, \ldots, n\}$ and $\kappa(G) \geq 1$ its vertex connectivity. Let $f(G)$ denote the number of free vertices of $G$ and $\mathrm{diam}(G)$ the diameter of $G$. The final goal of this paper is to determine all sequences of integers $(n,f,d,k)$ with $n\geq 8$, $f\geq 0$, $d\geq 2$ and $k\geq 1$ for which there exists a finite simple non-complete connected graph on $[n]$ with $f=f(G)$, $d=\mathrm{diam}(G)$ and $k=\kappa(G)$.
Local Existence and Conditional Regularity for the Navier-Stokes-Fourier System Driven by Inhomogeneous Boundary Conditions
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Abbatiello, Anna; Basarić, Danica; Chaudhuri, Nilasis; Feireisl, Eduard
We consider the Navier–Stokes–Fourier system with general inhomogeneous Dirichlet–Neumann boundary conditions. We propose a new approach to the local well-posedness problem based on conditional regularity estimates. By conditional regularity we mean that any strong solution belonging to a suitable class remains regular as long as its amplitude remains bounded. The result holds for general Dirichlet-Neumann boundary conditions provided the material derivative of the velocity field vanishes on the boundary of the physical domain. As a corollary of this result we obtain: Blow up criteria for strong solutions; Local existence of strong solutions in the optimal Lp - Lq framework; Alternative proof of the existing results on local well posedness.

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